One of the great concerns when tackling nonlinear systems is how to design a robust controller that is able to deal with uncertainty. Many researchers have been working on developing such type of controllers. One of the most efficient techniques employed to develop such controllers is sliding mode control (SMC). However, the low order SMC suffers from chattering problem which harm the actuators of the control system and thus unsuitable to be used in many practical applications. In this paper, the drawbacks of low order traditional sliding mode control (FOTSMC) are resolved by presenting a novel adaptive radial basis function neural network–based generalized ^{th} order sliding mode control strategy for n^{th} order uncertain nonlinear systems. The proposed solution adopts neural networks for their excellent capability in function approximation and thus used to approximate the nonlinearities and uncertainties for systems under consideration. The approximation errors are completely considered in the developed approach. The proposed approach can be used with any order of sliding mode and thus can be generally used with various types of applications. The global stability of the proposed control approach is proved through Lyapunov stability criterion. The proposed approach is validated and assessed through simulations on the nonlinear inverted pendulum system with severe modeling uncertainties. The simulations results show that the proposed approach provide superior performance compared with other approaches in the literature.

One of the most challenging problems while dealing with nonlinear systems is uncertainty [

Developing robust controllers for nonlinear systems is a great concern of many researchers over decades especially for real-time control systems [

Generally, the design of the SMC controller follows two main steps. The first step is the choice of a sliding surface on which the system trajectory is confined to lie and thus the parameters of this surface will govern the performance of the control system. The second step is to determine an efficient control law that forces the system states to reach the sliding surface from any initial state. In order to assure the system stability, the upper bound of uncertainties must be obtained precisely which cannot be guaranteed. Furthermore, chattering occurs due to the sign function in the overall control.

As a result of the efficiency of SMC in handling uncertainties and nonlinearities of systems, it has been the focus of many researchers over decades. SMC has been successfully applied in many applications. These applications include but are not restricted to space crafts [

Recently, many attempts have been seen trying to overcome the chattering problem while using the SMC methodology in nonlinear control systems [

Several second order SMC solutions are developed for nonlinear systems [

It has been apparent that working with higher order SMC is very efficient in overcoming the chattering problem which is the main concern of using SMC in many applications as it may cause high frequency oscillations which leads to actuators damage. However, as the order of SMC increases, the system model turns to be more complex. Developing the adaptive control laws for such complex systems is very challenging. Additionally, the time convergence of these systems cannot be guaranteed. This paper presents a novel adaptive generalized neuro-based SMC for nonlinear systems. The presented approach is of general order which means that we are able to deal with any order of SMC. The developed approach in this paper is supported with strong mathematical analysis and proofs. The main contributions of this work can be summarized as follows:

Developing a neuro-based SMC control approach for nonlinear systems which is capable of completely eliminating the chattering. The proposed approach can be used with any order of sliding mode and thus can be generally used with various types of applications. The neural networks are exploited in the proposed approach for their excellent capabilities in function approximation and thus used to approximate the nonlinearities and uncertainties in the system under consideration. The approximation errors are completely considered in the developed approach. All types of uncertainties have been taken into consideration.

The proposed approach is able to completely and adaptively estimate the uncertainties and thus the problems exist with either estimating the upper bound of the uncertainties (like in [

The stability of the designed system is validated using a suggested quadratic Lyapunov function. The estimated uncertainties are considered in the developed Lyapunov analysis. The developed control law guarantees that the system will reach the sliding surface in finite time with any order of sliding mode. The mathematical proof for the developed approach and the stability analysis are given in details below.

The proposed control algorithm is applied to a position trajectory tracking problem of an inverted pendulum nonlinear system through simulations. The proposed approach is compared to the existing conventional controllers. The simulation results indicate that the control performance of the develoepd control strategy is satisfactory and better than those of the existing conventional controllers.

The remainder of this paper is organized as follows: the problem under consideration is formulated in Section 2. Section 3 presents the proposed higher order sliding mode control for nonlinear systems. The presented adaptive neural network-based higher order sliding mode control is introduced in Section 4. Some simulation results and discussions are presented in Section 5. Finally, conclusions and some future directions are summarized in Section 6.

The ^{th} order uncertain nonlinear single-input single-output system may be described by:

where

In order to consider uncertainties in the system model, the controlled system can be modified as follows:

where

The sliding surface of order

where

In this section, a generalized extension of the authors' work in [^{th} order sliding mode control (DROSMC) for ^{th} order uncertain nonlinear single-input single-output system. In order to achieve that, the trajectory tracking error must approach zero and the ROSMC control law must steer the sliding surface

where

The time derivative of sliding surface in

The ^{th} time derivative of sliding surface is computed as:

Substituting

In order to keep the error variables of the nominal system

The switching control signal

where

where

By differentiating

Substituting from

Substituting from

Substituting

By using

In order to ensure that the closed loop system is globally stable and the reaching condition

The total control law of the developed DROSMC controller can be deduced by substituting from

The block diagram of the proposed DROSMC is shown in

Although, the DROSMC controller described in ^{th} order sliding mode control strategy (ARBFNN-CROSMC). The developed continuous strategy is based on replacing the discontinuous term with a continuous control action. The approximation property of the RBFNN is used to approximate the unknown nonlinear functions

where

where ^{th} Gaussian function.

The adaptive RBFNN continuous ^{th} order sliding mode control strategy (ARBFNN-CROSMC) is chosen as:

where

where

where

The control signal

From

Substituting

Using

Substituting

where

Define

Substituting

The adaptive control term

where

where

The first derivative of Lyapunov function

Substituting from

Substituting from

Rearranging

In order to ensure that the closed loop system is globally stable and the reaching condition

The time evolution of the estimated uncertainty

Substituting

For a small positive number

From the above analysis, the global asymptotic stability of the closed loop system is guaranteed.

In order to validate the effectiveness of the proposed ARBFNN-CROSMC control methodology, the stabilization control of the inverted pendulum (IP) system [

First, a discontinuous second order sliding mode control approach (D2SMC) and an adaptive RBFNN continuous second order sliding mode control strategy (ARBFNN-C2SMC) are implemented using the inverted pendulum system. The performance of both conventional PID controller and D2SMC are compared to demonstrate the efficacy of the proposed ARBFNN-C2SMC control strategy. Two main problems are implemented; the set point tracking control problem, and the trajectory tracking control problem.

To check the ability of the proposed strategy to reach a desired reference point, a desired angular position is set at

Algorithm | |||
---|---|---|---|

PID | 309.3860 | 100.429 | 5.5939 |

D2SMC | 120.8765 | 55.4535 | 3.6689 |

ARBFNN-C2SMC | 21.4102 | 15.9799 | 2.0412 |

In this subsection, the performance of the proposed approach is assessed in case of trajectory tracking control problem. The standalone PID parameters are set as:

Again, the simulation results illustrate the superiority of the proposed control approach compared with other approaches in case of trajectory tracking control problem. As seen, the D2SMC controller has severe chattering while the proposed approach is able to completely eliminate any type of chattering. A comparison of tracking response characteristics are summarized in

Algorithm | |||
---|---|---|---|

PID | 1054.3 | 1583.8 | 749.314 |

D2SMC | 118.6571 | 20.8729 | 0.5459 |

ARBFNN-C2SMC | 34.9895 | 5.3656 | 0.0089 |

Second, the performance of the proposed approach is assessed in case of using third order sliding mode control (

The reference desired angular position is set as:

As seen in

Algorithm | |||
---|---|---|---|

PID | 309.3860 | 100.429 | 5.5939 |

D3SMC | 105.3567 | 50.9653 | 2.4885 |

ARBFNN-C3SMC | 15.4102 | 10.3421 | 1.0982 |

In this subsection, the performance of the proposed approach is assessed in case of trajectory tracking control problem. A step disturbance force of

Again, the simulation results illustrate the superiority of the proposed control approach compared with other approaches in case of trajectory tracking control problem. As seen, the D3SMC controller has severe chattering while the proposed approach is able to completely eliminate any type of chattering. A comparison of tracking response characteristics are summarized in

Algorithm | |||
---|---|---|---|

PID | 1054.3 | 1583.8 | 749.314 |

D3SMC | 116.6991 | 18.3527 | 0.3769 |

ARBFNN-C3SMC | 31.5485 | 3.9646 | 0.0067 |

An adaptive radial basis function neural network-based generalized ^{th} order sliding mode control approach for ^{th} order uncertain nonlinear systems is developed in this paper. The proposed approach is able to deal with severe uncertainties even for high order nonlinear systems. The proposed approach employed the radial basis function neural networks for their efficiency in approximating nonlinear functions. The proposed approach has been validated theoretically for generalized ^{th} order sliding mode control. The simulation results showed that the proposed approach has great performance compared with other approaches.

The authors extend their appreciation to the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number (IF-PSAU-2021/01/17796).